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# list rational zeros?

Given f(x)=x^3+2x^2-13x+10, (a) list all possible rational zeros, (b) find all rational zeros, and (c) factor f(x).

### 2 Answers by Expert Tutors

Robert J. | Certified High School AP Calculus and Physics TeacherCertified High School AP Calculus and Ph...
4.6 4.6 (13 lesson ratings) (13)
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a) Rational zeros = ±factors of 10 = ±1, ±2, ±5, ±10

b) f(-5) = f(1) = f(2) = 0
So, rational zeros are x = -5, 1, 2

c) f(x) = (x+5)(x-1)(x-2)
Arthur D. | Effective Mathematics TutorEffective Mathematics Tutor
5.0 5.0 (7 lesson ratings) (7)
0
using synthetic division, possible zeroes are factors of 10 divided by factors of 1, the coefficient
of x^3, so we have +1, +2, +5, and +10

let's try 1

1             1    2    -13    10
1       3   -10
1    3    -10     0

now divide x^3+2x^2-13x+10 by x-1

x^2 + 3x - 10
________________
(x-1)   / x^3+2x^2-13x+10
/   x^3-x^2
_________
3x^2-13x
3x^2-3x
_________
-10x+10
-10x+10
________
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now factor x^2+3x-10 into (x+5)(x-2)
so we have (x-1)(x+5)(x-2) for the factorization and 1, -5, and 2 for the zeroes